Heretofore, sine waves have been generated with digital circuitry either by direct read-out of values stored in a memory unit or by a digital signal processor generating the sequential values of the sine wave using a recursive process. One known method calculates successive samples of a pair of orthogonal sine waves (i.e., a sine wave and a cosine wave) by means of recursion equations. Each successive value, y(n) and z(n) of the sine and cosine wave can be calculated from the previous values, y(n-1) and z(n-1), using the well-known trigonometric formulae: EQU y(n)=y(n-1) cos .omega.+z(n-1) sin .omega. (1) EQU z(n)=z(n-1) cos .omega.-y(n-1) sin .omega. (2)
where .omega.=2.pi.FT, T is the interval between successive samples and F is the frequency of the sine wave. The calculating procedure suffers from the accumulation of round-off errors leading to the generation of an exponentially increasing or decreasing sine wave.
U.S. Pat. No. 4,285,044, L. Thomas et al., issued Aug. 18, 1981, proposed that round-off error accumulation could be moderated by the use of rather elaborate circuitry for calculating the quantity ##EQU2## and using the calculated quantity as an approximation for a normalization factor to be applied to calculated sample values to ensure that the value of the z.sup.2 (n)+y.sup.2 (n) calculated from sample values of the sine and cosine waves would initially converge to 1 and subsequently remain at approximately 1. The Thomas patent normalizing factor circuitry prevents exponential build up or exponential decay of the calculated values of the sine and cosine waves. A less complex arrangement for avoiding the exponential build up or decay problem in a system for generating successive sample values of a sine wave in digital format would clearly be attractive. It would also be advantageous to permit a given digital signal processor system to generate the sample values for more sine waves or generate more frequent samples of a sine wave and thereby achieve a more perfect output wave when an analog sine wave is generated by a digital to analog converter and a filter at the output of that converter is not perfect.
U.S. Pat. No. 4,577,287 by C. Chrin, issued Mar. 18, 1986, proposed that roundoff error accumulation could be moderated by the use of circuitry or calculations which limit the magnitudes of calculated values of sine and cosine to one. Initial constants would be rounded up so that successive calculated values would tend to build up exponentially very slowly and be limited by the limiting step. By biasing the initial constants exponential decay is avoided. However, the solution proposed by Chrin suffers from the fact that a large number of calculating steps are required for calculating each sample, and that, for many frequencies, the amplitude of the desired output varies by several percent over an extended interval of time.
A problem of the prior art is therefore that the number of computation steps for computing successive values of samples of a sine wave is high and that the amount of computation for stabilizing the magnitude of such samples is high.